Time-changed fractional Ornstein-Uhlenbeck process

Ascione, Giacomo; Mishura, Yuliya; Pirozzi, Enrica

doi:10.1515/fca-2020-0022

Cited by 14 publications

(10 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The lack of such sufficient condition can also be seen from the phase portrait in Figure 3, as the maximum is reached in a region in which d Φ I Φ dt Φ is still strictly positive. Let us indeed recall that the classical Fermat theorem on extremal points (see, e.g., [60] Theorem 5.9) becomes an inequality in the non-local context (see, e.g., [61] Theorem 1 in the fractional case or [62] Proposition 2.2 in the general Caputo-type case), justifying the fact that, after the function reaches a maximum and starts decreasing, the non-local derivative could still be non-negative. Finally, let us observe that the lack of semigroup property can be overtaken by considering a deformation map θ, as shown in Appendix A, that takes in consideration the fact that the system remembers all its history.…”

Section: Discussionmentioning

confidence: 99%

On the Construction of Some Deterministic and Stochastic Non-Local SIR Models

Ascione

2020

Mathematics

Self Cite

View full text Add to dashboard Cite

Fractional-order epidemic models have become widely studied in the literature. Here, we consider the generalization of a simple SIR model in the context of generalized fractional calculus and we study the main features of such model. Moreover, we construct semi-Markov stochastic epidemic models by using time changed continuous time Markov chains, where the parent process is the stochastic analog of a simple SIR epidemic. In particular, we show that, differently from what happens in the classic case, the deterministic model does not coincide with the large population limit of the stochastic one. This loss of fluid limit is then stressed in terms of numerical examples.

show abstract

Section: Discussionmentioning

confidence: 99%

On the Construction of Some Deterministic and Stochastic Non-Local SIR Models

Ascione

2020

Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…where D h t is a generalized time-fractional derivative of Caputo type, which is defined (for sufficiently good functions v : (0, ∞) → R of time variable t) via the Laplace transform (cf. [1]) by…”

Section: Corollarymentioning

confidence: 99%

Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations

Bender

Bormann

Butko

2022

Fract Calc Appl Anal

View full text Add to dashboard Cite

We consider a class of generalized time-fractional evolution equations containing a fairly general memory kernel k and an operator L being the generator of a strongly continuous semigroup. We show that a subordination principle holds for such evolution equations and obtain Feynman-Kac formulae for solutions of these equations with the use of different stochastic processes, such as subordinate Markov processes and randomly scaled Gaussian processes. In particular, we obtain some Feynman-Kac formulae with generalized grey Brownian motion and other related self-similar processes with stationary increments.

show abstract

“…Here − − 1 2 ∆ γ , γ ∈ (0, 1), is the fractional Laplacian [30]. Equation (1) serves as a governing equation for the process (Y E β t ) t⩾0 which is a symmetric 2γ-stable Lévy process (Y t ) t⩾0 time-changed by an independent inverse β-stable subordinator (E β t ) t⩾0 .…”

Section: Introductionmentioning

confidence: 99%

“…The governing equation of GGBM is the following time-stretched time-fractional heat equation Equation (3) reduces to the time-fractional heat equation (i.e. equation (1) with γ ∶= 1) if α ∶= β. For γ ∈ (0, 1), the time-and space-fractional heat equation ( 1) is shown to be the governing equation for another SSSI RSGP (see [45]).…”

Section: Introductionmentioning

confidence: 99%

Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations

Bender,

Bormann,

Butko

2022

Preprint

View full text Add to dashboard Cite

We consider generalized time-fractional evolution equations of the formwith a fairly general memory kernel k and an operator L being the generator of a strongly continuous semigroup. In particular, L may be the generator L 0 of a Markov process ξ on some state space Q, or L ∶= L 0 + b∇ + V for a suitable potential V and drift b, or L generating subordinate semigroups or Schrödinger type groups. This class of evolution equations includes in particular time-and space-fractional heat and Schrödinger type equations. We show that a subordination principle holds for such evolution equations and obtain Feynman-Kac formulae for solutions of these equations with the use of different stochastic processes, such as subordinate Markov processes and randomly scaled Gaussian processes. In particular, we obtain some Feynman-Kac formulae with generalized grey Brownian motion and other related self-similar processes with stationary increments.

show abstract

Time-changed fractional Ornstein-Uhlenbeck process

Cited by 14 publications

References 49 publications

On the Construction of Some Deterministic and Stochastic Non-Local SIR Models

On the Construction of Some Deterministic and Stochastic Non-Local SIR Models

Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations

Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations

Contact Info

Product

Resources

About